# Definition:Calculus/Integral

## Definition

**Integral calculus** is a subfield of calculus which is concerned with the study of the rates at which quantities accumulate.

Equivalently, given the rate of change of a quantity **integral calculus** provides techniques of providing the quantity itself.

The equivalence of the two uses are demonstrated in the Fundamental Theorem of Calculus.

The technique is also frequently used for the purpose of calculating areas and volumes of curved geometric figures.

## Also see

- Results about
**integral calculus**can be found here.

## Linguistic Note

The term **integral calculus** is the Anglified version of the neo-Latin phrase **calculus integralis**, made popular by Gottfried Wilhelm von Leibniz.

At first he suggested **calculus summatorius**, but in $1696$ he decided with Johann Bernoulli that **calculus integralis** would be better.

Some sources suggest that the name **integral**, first used by him in $1690$, may well have been coined by Jacob Bernoulli.

## Historical Note

The techniques used by Archimedes of Syracuse to calculate areas and volumes of curvilinear figures are often suggested as being the precursors to **integral calculus**.

Much of the early work developing **integral calculus** was done by Isaac Newton.

His initial work on this seems to have been achieved during the years $1665$ to $1667$ when he was at home in Woolsthorpe.

At the same time that Newton was arranging his thesis, Gottfried Wilhelm von Leibniz was publishing many papers himself on the same subject.

The rigorous treatment of the subject was developed later, by Carl Friedrich Gauss, Niels Henrik Abel and Augustin Louis Cauchy.

## Sources

- 1937: Eric Temple Bell:
*Men of Mathematics*... (previous) ... (next): Chapter $\text{II}$: Modern Minds in Ancient Bodies - 1937: Eric Temple Bell:
*Men of Mathematics*... (previous) ... (next): Chapter $\text{VI}$: On the Seashore - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $8$: The System of the World: Calculus